For the simplicity, we set \(\mathcal {L}_a^{min} := \min _{\theta \in \mathbb {R}^d} \mathcal {L}_a(\theta ) = 0\)  and \(\mathcal {L}_b^{min} := \min _{\theta \in \mathbb {R}^d} \mathcal {L}_b(\theta ) = 0\)  in our proofs without loss of generality, which is a common property observed in training deep neural networks [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}.
